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Find the Image of a Point Having the Position Vector: `3hati - 2hatj + Hat K` in the Plane `Vec R.(3hati - Hat J + 4hatk) = 2` - Mathematics

Find the image of a point having the position vector: `3hati - 2hatj + hat k` in the plane `vec r.(3hati - hat j + 4hatk) = 2`

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Solution

Let B be the root of point `A(3hati - 2hatj + hatk)` in the plane `vec r(3hati - hatj + 4hatj) = 2`can of AB: is

`barr(3hati - 2hatj + hatk) + lambda(3hati - hatj + 4hatk)`

`:. (x - 3)/3 = (y + 2)/(-1) = (z-1)/4 = lambda`

`:. x = 3lambda + 3`,  `y = -lambda -2`, `z = 4lambda + 1`

subtitute x,y and z in plane 3x - y + 4z = 2

`∴ 3(3lambda + 3)-(-lambda - 2) + 4(4lambda + 1) = 2`

`9lambda + 9 + lambda + 2 + 16lambda + 4 = 2`

`26lambda + 13 = 0 => lambda = -1/2`

`:. x = -3/2 + 3`, `y = 1/2 - 2` `z = -2 + 1`

x = 3/2, `y = -3/2`, z = -1

∴ by midpt formula

`3/2 = (3+x_1)/2, (-3)/2 = (-2+y1)/z, -1 = (1-z_1)/2`

`x_1 = 0`  `y_1 = -1`, `-z = 1 - z_1`

`z_1 = 1+ 2 = 3`

Concept: Vector and Cartesian Equation of a Plane
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